Generalized M-polynomial and graphical invariants of silicate cage networks with applications to property prediction

Zhang Yan; Muhammad Salman; Syed Shahzaib; Imran Khalid; Hanen Karamti; Zhang Yan; Muhammad Salman; Syed Shahzaib; Imran Khalid; Hanen Karamti

doi:10.3934/math.2026880

AIMS Mathematics

2026, Volume 11, Issue 7: 21743-21780. doi: 10.3934/math.2026880

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Generalized M-polynomial and graphical invariants of silicate cage networks with applications to property prediction

1.
School of Information Engineering, Anhui International Business Vocational College, Hefei, China
2.
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
3.
Center for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan
4.
Department of Computer Sciences, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 20 May 2026 Revised: 08 July 2026 Accepted: 10 July 2026 Published: 22 July 2026
MSC : 05C10, 68R05, 68R10, 68T01

This study introduces a generalized M-polynomial framework for the structural analysis of silicate cage molecular networks. From the molecular graph representation, several degree-based topological indices are derived, including Zagreb indices, the Randić index, the inverse Randić index, the symmetric division index, the harmonic index, the sum inverse index, and the augmented Zagreb index. The proposed formulation incorporates degree-weighted parameters, enabling a systematic characterization of structural complexity in terms of network parameters such as the number of layers and replicated units. Case-wise analysis for different structural configurations shows consistent trends in all indices, reflecting their sensitivity to molecular growth and connectivity. Overall, the results demonstrate that the proposed framework provides a unified and scalable approach for analyzing silicate cage networks. The obtained indices effectively capture structural variations and offer useful insights for graph-theoretical studies of complex molecular systems.
- silicate cage networks,
- generalized M-polynomial,
- topological indices,
- graph theory,
- molecular graph invariants
Citation: Zhang Yan, Muhammad Salman, Syed Shahzaib, Imran Khalid, Hanen Karamti. Generalized M-polynomial and graphical invariants of silicate cage networks with applications to property prediction[J]. AIMS Mathematics, 2026, 11(7): 21743-21780. doi: 10.3934/math.2026880

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Abstract

This study introduces a generalized M-polynomial framework for the structural analysis of silicate cage molecular networks. From the molecular graph representation, several degree-based topological indices are derived, including Zagreb indices, the Randić index, the inverse Randić index, the symmetric division index, the harmonic index, the sum inverse index, and the augmented Zagreb index. The proposed formulation incorporates degree-weighted parameters, enabling a systematic characterization of structural complexity in terms of network parameters such as the number of layers and replicated units. Case-wise analysis for different structural configurations shows consistent trends in all indices, reflecting their sensitivity to molecular growth and connectivity. Overall, the results demonstrate that the proposed framework provides a unified and scalable approach for analyzing silicate cage networks. The obtained indices effectively capture structural variations and offer useful insights for graph-theoretical studies of complex molecular systems.

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